

To implement these schemes, we used the Firedrake software, [1]}, which is a library for solving PDEs using finite element methods and is built on the PETSc solver library [2]}.
Firedrake constructs the quadrilateral Raviart-Thomas elements as tensor-product elements [3]} and provides support for the hybridised solver [4]} used in the shallow-water model of Section REF . The orthographic projections were plotted using the Cartopy python package [5]}. Finally, for the SUPG method used in the stabilised voriticity discretisation, we consider a stabilisation parameter of the form
\(\tau = \left(\lambda \frac{2}{\Delta t} + \frac{2|\mathbf {u}|}{\Delta x} \right)^{-1},\) 
